| L(s) = 1 | + (7.86 − 7.86i)3-s + (−27.2 + 27.2i)5-s + 50.3·7-s − 42.8i·9-s + (53.1 + 53.1i)11-s + (125. + 125. i)13-s + 428. i·15-s + 286.·17-s + (99.5 − 99.5i)19-s + (395. − 395. i)21-s − 100.·23-s − 858. i·25-s + (300. + 300. i)27-s + (−343. − 343. i)29-s − 208. i·31-s + ⋯ |
| L(s) = 1 | + (0.874 − 0.874i)3-s + (−1.08 + 1.08i)5-s + 1.02·7-s − 0.528i·9-s + (0.438 + 0.438i)11-s + (0.740 + 0.740i)13-s + 1.90i·15-s + 0.990·17-s + (0.275 − 0.275i)19-s + (0.897 − 0.897i)21-s − 0.189·23-s − 1.37i·25-s + (0.412 + 0.412i)27-s + (−0.408 − 0.408i)29-s − 0.216i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.27750 + 0.313204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.27750 + 0.313204i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-7.86 + 7.86i)T - 81iT^{2} \) |
| 5 | \( 1 + (27.2 - 27.2i)T - 625iT^{2} \) |
| 7 | \( 1 - 50.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + (-53.1 - 53.1i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (-125. - 125. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 - 286.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (-99.5 + 99.5i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + 100.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (343. + 343. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + 208. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (-1.15e3 + 1.15e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 2.33e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-2.07e3 - 2.07e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 1.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (2.13e3 - 2.13e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (3.72e3 + 3.72e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (2.49e3 + 2.49e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (-329. + 329. i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 1.04e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.67e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 4.47e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (1.45e3 - 1.45e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 1.14e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.31e4T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.67110585767264150551341597563, −11.55889263481542164305568208906, −10.98319509838007974019715513571, −9.312678928896310647958599017382, −7.84459903268413970253516108125, −7.70879105920290930392518146609, −6.42663011521799468072968649745, −4.31766659002725372850071687217, −3.00319673909414750526190464228, −1.52648888863007625973512056726,
1.05109660725596742076604051965, 3.42389428617203386380743228178, 4.27831375093785990041506684190, 5.47046716283818933730347659900, 7.74702028698925001548290022571, 8.403353208917993196833678672800, 9.127309200116131422096478597687, 10.48658896657570984053053469046, 11.61047937893821554236317372613, 12.46076993960826290331928403068
